The maximum agreement subtree problem

In this paper, we investigate an extremal problem on binary phylogenetic trees. Given two such trees T-1 and T-2, both with leaf-set {1, 2, . . . , n}, we are interested in the size of the largest subset S subset of {1, 2, . . . , n} of leaves in a common subtree of T-1 and T-2. We show that any two binary phylogenetic trees have a common subtree on Omega (root log n) leaves, thus improving on the previously known bound of Omega (log log n) due to Steel and Szekely. To achieve this improved bound, we first consider two special cases of the problem: when one of the trees is balanced or a caterpillar, we show that the largest common subtree has Omega (log n) leaves. We then handle the general case by proving and applying a Ramsey-type result: that every binary tree contains either a large balanced subtree or a large caterpillar. We also show that there are constants c, alpha > 0 such that, when both trees are balanced, they have a common subtree on cn(alpha) leaves. We conjecture that it is possible to take alpha = 1/2 in the unrooted case, and both c = 1 and alpha = 1/2 in the rooted case. (c) 2013 Elsevier B.V. All rights reserved.